Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

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Representation: A relation can be represented as a directed graph (or digraph). For (a,b),R, a and b are vertices (or nodes) in the graph and a directional edge runs from a to b. Example: The following digraph represents {(a,b), (b,c), (c,a), (c,b)}. a" " b " c What about all of those examples on page 159 of the class notes? We can do all of them over in either representation. 163 Examples (from page 159): ! # 1 1 0 0 # • M R1 = # 1 1 0 0 # # 0 0 0 1 # 1 0 0 1 "# ! # # # # # # "# ! # # # # # # "# $ & & & & & & %& $ & & & & & & %& $ & & & & & & %& 1 1 0 0 • M R2 = 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 • M R3 = 1 1 0 0 0 0 1 0 1 0 0 1 or a digraph a 1 " " a 2 164
! # 0 0 0 0 # • M R4 = # 1 0 0 0 # # 1 1 0 0 # 1 1 1 0 "# ! # # # # # # "# ! # # # # # # "# 1 1 1 1 • M R5 = 1 1 1 1 0 0 1 0 1 0 0 1 0 0 0 0 • M R6 = 0 0 0 0 0 0 0 1 0 0 0 0 $ & & & & & & %& $ & & & & & & %& $ & & & & & & %& or the digraph a 3 " " a 4 165 Definition: A relation on a set A is an equivalence relation if it is reflexive, symmetric, and transitive. Two elements a and b that are related by an equivalence relation are called equivalent and denoted a~b. Examples: • Let A = Z. Define aRb if and only if either a = b or a = 4b. o symmetric: aRa since a = a. o reflexive: aRb E bRa since a = ±b. o transitive: aRb and bRc E aRc since a = ±b = ±c. • Let A = R. Define aRb if and only if a4b,Z. o symmetric: aRa since a4a = 0,Z. o reflexive: aRb E bRa since a4b,Z E 4(a4b) = b4a,Z. o transitive: aRb and bRc E aRc since (a4b)+(b4c) ,Z E a4c,Z. 166
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Discrete Mathematics University of
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Definition: A proposition is a stat
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We can compound logical operators t
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Theorem: p" (q!r) ' (p"q)!(p"r). Pr
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Propositional logic is pretty limit
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Rules of Inference are used instead
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Sets, Functions, Sequences, and Sum
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Definition: n sets A i are disjoint
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Definition: The composition of n fu
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Some other common summations with c
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Notation: Timing, as a function of
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! $ procedure binary_search(x, " a
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Definition: If a,b,Z and a-0, we sa
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Theorem: There are infinitely many
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Algorithm: Addition of integers pro
Page 31 and 32: Euclidean Algorithm: Compute gcd(a,
Page 33 and 34: Sun Tzu’s Puzzle: The a k ,{2, 1,
Page 35 and 36: • Sometimes P(k) is for a (possib
Page 37 and 38: Example: Every postage amount < $.1
Page 39 and 40: Definition: A recursive algorithm s
Page 41 and 42: Merge sort is a balanced binary tre
Page 43 and 44: Examples: • Consider 3 students i
Page 45 and 46: Theorem: The number of r-combinatio
Page 47 and 48: If we allow repetitions in the perm
Page 49 and 50: Algorithm: Generating the next r-co
Page 51 and 52: Theorem: Let E and F be events in a
Page 53 and 54: Note: For bit strings of length 4,
Page 55 and 56: Theorem: If X i , 1;i;n, are random
Page 57 and 58: Definition: The random variables X
Page 59 and 60: Example: We have 2 boxes. The first
Page 61 and 62: Let E be the event that an incoming
Page 63 and 64: Advanced Counting Principles Defini
Page 65 and 66: Examples: Typical ones include •
Page 67 and 68: Now comes the second case for n = 2
Page 69 and 70: Theorem: Assume {b i },{c i },R. Su
Page 71 and 72: Example (funny integer multiplicati
Page 73 and 74: • In step 4, we have to take care
Page 75 and 76: Definition: The extended binomial c
Page 77 and 78: Example: a n = 8a n41 + 10 n41 with
Page 79 and 80: Relations Definition: A relation on
Page 81: Representation: The relation R from
Page 85 and 86: Definition: A partition of a set S
Page 87 and 88: Examples of Simple Graphs: • A co
Page 89 and 90: Examples: v 1 v 2 • and v 4 v 3 v
Page 91 and 92: Theorem: Let G = (V,E) be a graph w
Page 93 and 94: Definition: A coloring of a simple
Page 95 and 96: Definition: A m-ary tree is a roote
Page 97 and 98: Definition: A decision tree is a ro
Page 99 and 100: Note: Game trees are another highly
Page 101 and 102: Definition (Postorder Traversal): L
Page 103 and 104: Definition: Let G = (V,E) be a simp
Page 105 and 106: Definition: A minimum spanning tree
Page 107 and 108: Examples: Most circuits are of the
Page 109: Definition: An implicant is sum ter
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